I think Gladio is an interesting character. On one hand, it's his job to be a literal human shield to Noctis. So his life by definition is less important than Noct's. And he knows for sure that his father, the king's Shield, died trying to protect King Regis. It's a duty he takes seriously.

On the other hand, he's a hothead and lacks patience.

So to see Noctis fall apart like that, he must have thought to himself "am I really going to put my life on the line for this guy"? Someone who seemingly doesn't care enough about the lives that have been sacrificed for him so far, who wilts under pressure rather than march forward to get revenge/face his destiny, etc?

But as far as his character in relation to the prompt, I can see him being really impatient with Noct up until the point he gets the pertinent information that Noctis has an old injury. And then he'd probably work around the problem with him. (And then if Noct continued to ignore the stretching exercises and physical therapy and continued to do badly, he's probably get annoyed with him again, yeah)